Finite volume methods for solid mechanics are attractive for multiphysics applications due to their local conservation properties and natural compatibility with fluid solvers, yet they are still predominantly solved using segregated or explicit algorithms, which are not robust and exhibit slow nonlinear convergence. In this contribution, we present a Jacobian-free Newton– Krylov (JFNK) solution strategy for cell-centred finite-volume solid mechanics and demonstrate how it enables efficient, Newton-type convergence without explicit Jacobian assembly, even for higher-order discretisations. The proposed approach reformulates the finite-volume solid-mechanics equations in residual form and applies a Newton–Krylov method, in which Jacobian–vector products are approximated by finite differencing the residual. A compact-stencil approximate Jacobian, identical to that used in conventional segregated finite-volume solvers, is employed as a preconditioner for Krylov iterations. This choice allows the method to be integrated into existing finite-volume frameworks (e.g., OpenFOAM) with minimal modification, while retaining favourable memory characteristics. We first assess the performance of the JFNK approach for second-order cell-centred finite volume formulations across a range of static, dynamic, linear, and nonlinear benchmark problems. Results demonstrate substantial improvements in convergence rate and computational efficiency relative to traditional segregated algorithms, with order-of- magnitude speed-ups observed for many cases. We then extend the framework to third- and fourth-order finite discretisations based on least-squares gradient reconstruction and Gaussian quadrature at cell faces. Despite the increased complexity of the discretisation, the same compact-stencil preconditioner remains effective, enabling robust and efficient solution of higher-order finite-volume systems. The results indicate that Jacobian-free Newton–Krylov methods provide a powerful and practical route to bringing fully implicit, high-order solution strategies to finite-volume solid mechanics, bridging a long-standing gap with finite-element methodologies and opening new opportunities for accurate, scalable multiphysics simulation.